Cu-based I–III–VI2 materials have enjoyed much attention as candidate solar cell adsorbers. While the vast majority of studies has centered on materials with group 13 (In, Ga) as the trivalent metal, the scarcity and expense of In has motivated a research drive to discover alternative Cu-based absorber materials. In this study, we use screened hybrid density functional theory (DFT) to investigate the electronic structure and bonding in some novel I–III–VI2 materials, namely, CuMCh2 (M = Sb, Bi; Ch = S, Se). We demonstrate that these materials possess fundamental band gaps that are indirect in nature, which is at variance with previous experimental results. We analyze the crystal structures and rationalize the structural differences between these and typical chalcopyrite materials. The band structure features and bonding of these materials are then discussed in relation to their utility as solar cell absorbers.